Description Usage Arguments Details Value Author(s) See Also Examples

This function computes an equivalent nonexceedance probability *F* of a single value *x* for the sample data set (*\hat{X}*) through inversion of the empricial quantile function as computable through Bernstein or Kantorovich Polynomials by the `dat2bernqua`

function.

1 | ```
dat2bernquaf(x, data, interval=NA, ...)
``` |

`x` |
A scalar value for which the equivalent nonexceedance probability |

`data` |
A vector of data values that directly correspond to the argument |

`interval` |
The search interval. If |

`...` |
Additional arguments passed to |

The basic logic is thus. The *\hat{X}* in conjunction with the settings for the polynomials provides the empirical quantile function (EQF). The `dat2bernquaf`

function then takes the EQF (through dynamic recomputation) and seeks a root for the single value also given.

The critical piece likely is the search interval, which can be modified by the `interval`

argument if the internal defaults are not sufficient. The default interval is determined as the first and last Weibull plotting positions of *\hat{X}* having a sample size *n*: *[1/(n+1), 1 - 1/(n+1)]*. Because the `dat2bernqua`

function has a substantial set of options that control how the empirical curve is (might be) extrapolated beyond the range of *\hat{X}*, it is difficult to determine an always suitable interval for the rooting. However, it should be considered obvious that the result is more of an interpolation if *F(x)* is within *F \in [1/(n+1), 1 - 1/(n+1)]* and increasingly becomes an accurate interpolation as *F(x) \rightarrow 1/2* (the median).

If the value *x* is too far beyond the data or if the search interval is not sufficient then the following error will be triggered:

1 2 |

The Examples section explores this aspect.

An **R** `list`

is returned.

`x` |
An echoing of the |

`f` |
The equivalent nonexceedance probability |

`interval` |
The search interval of |

`afunc.root` |
Corresponds to the |

`iter` |
Corresponds to the |

`estim.prec` |
Corresponds to the |

`source` |
An attribute identifying the computational source: “dat2bernquaf” |

W.H. Asquith

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 | ```
dat2bernquaf(6, c(2,10)) # median 1/2 of 2 and 10 is 6 (trivial and fast)
## Not run:
set.seed(5135)
lmr <- vec2lmom(c(1000, 400, 0.2, 0.3, 0.045))
par <- lmom2par(lmr, type="wak")
Q <- rlmomco(83, par) # n = 83 and extremely non-Normal data
lgQ <- max(Q) # 5551.052 by theory
dat2bernquaf(median(Q), Q)$f # returns 0.5100523 (nearly 1/2)
dat2bernquaf(lgQ, Q)$f # unable to root
dat2bernquaf(lgQ, Q, bound.type="sd")$f # unable to root
itf <- c(0.5, 0.99999)
f <- dat2bernquaf(lgQ, Q, interval=itf, bound.type="sd")$f
print(f) # F=0.9961118
qlmomco(f, par) # 5045.784 for the estimate F=0.9961118
# If we were not using the maximum and something more near the center of the
# distribution then that estimate would be closer to qlmomco(f, par).
# You might consider lqQ <- qlmomco(0.99, Q) # theoretical 99th percentile and
# let the random seed wander and see the various results.
## End(Not run)
``` |

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